Root Mean Square Calculator (RMS / quadratic mean of a data set)

Understanding the root mean square calculator — concept and principle

This calculator finds the root mean square (RMS), also called the quadratic mean, of a list of numbers. It also shows the sum of squares, the mean of the squares, the arithmetic mean, the mean of the absolute values, the population standard deviation and a check. Use it for the effective value of AC voltages and currents, for vibration and noise levels, for error measures such as RMSE, and for statistics homework.

Squaring makes −2 count by its size; the root of the mean square is 4.24, above the plain mean 3.2.

How it is calculated

x_RMS = √((x₁² + x₂² + … + xₙ²) ÷ n)

Square every value.
Take the mean of the squares.
Take the square root.

Check: x_RMS = √(mean² + σ²), where σ is the population standard deviation (divided by n).

The principle

Squaring removes the sign, so negative and positive values both count by their size, and large values count more than small ones. The square root at the end brings the result back to the original units. That makes the RMS the right "typical size" for quantities whose effect depends on the square, such as electrical power (P = V²/R) or energy in a wave. The identity RMS² = mean² + σ² shows that the RMS combines the average level and the spread around it. It also shows that the RMS is never smaller than the arithmetic mean or the mean absolute value.

Worked example

Default values: 3, 4, 5, −2, 6.

Squares: 9, 16, 25, 4, 36; sum = 90.
Mean of the squares: 90 ÷ 5 = 18.
RMS = √18 ≈ 4.242640687.
Arithmetic mean = 16 ÷ 5 = 3.2; mean of absolute values = 20 ÷ 5 = 4.
σ = √(18 − 3.2²) = √7.76 ≈ 2.785677655; check √(3.2² + 7.76) = √18.

How to read the result

The RMS is always at least as large as the mean of absolute values; the gap grows when the values are uneven or contain large outliers. For a pure sine wave, RMS = peak ÷ √2 ≈ 0.707 × peak, which is why a 230 V mains supply has a peak of about 325 V. When the RMS is used as an error measure, a value close to 0 means predictions are close to the observations.

Common mistakes

Taking the square root of each value instead of the root of the mean of squares.
Dividing by n − 1 instead of n.
Using the arithmetic mean of a signal that swings positive and negative; it can be near 0 even when the signal is large.
Applying the peak ÷ √2 rule to waveforms that are not pure sine waves.

Keep in mind: For electrical measurements, a meter labelled "true RMS" measures the RMS of any waveform; cheaper meters may assume a sine wave and read incorrectly on distorted signals. Follow safety rules when working with mains voltage.

Frequently asked questions

What is the difference between RMS and average?

The average lets positive and negative values cancel; the RMS measures size regardless of sign and weights large values more.

Why is mains voltage quoted as RMS?

Because the RMS gives the same heating power in a resistor as a steady DC voltage of that value.

Is RMS the same as standard deviation?

Only when the mean is 0. In general RMS² = mean² + σ².

Sources: Weisstein, E. W., "Root-Mean-Square", Wolfram MathWorld

Checked 9 Nov 2026. Weisstein, E. W., “Root-Mean-Square”, Wolfram MathWorld (x_RMS = √((x₁² + … + xₙ²)/n), eqs. 1–3; the RMS is the power mean M₂). Identity RMS² = mean² + σ² (population variance) — standard result.

Your numbers